\chapter{Advanced Analytical Mechanics}

The Hamilton-Jacobi (HJ) formulation is the final reformulation of classical mechanics, expressing the entire dynamics of a system as a single first-order partial differential equation for a scalar function $S$, called the \textbf{principal function}. Solving the HJ equation by separation of variables often yields complete solutions more directly than the Lagrange or Hamilton equations -- especially for systems with symmetries and cyclic coordinates. The HJ framework also provides the classical foundation for the WKB approximation and connects to the Schrodinger equation in the $\hbar \to 0$ limit.

This chapter is organized in three parts. Section 3.1 develops the HJ equation from Hamiltonian mechanics and introduces separation of variables, action-angle variables, and electromagnetic minimal coupling. Section 3.2 applies the HJ formalism to classical mechanics problems: the free particle, projectile motion, the simple harmonic oscillator, the Kepler (two-body) problem, and the rigid rotator on a sphere. Section 3.3 treats problems from electromagnetism, including charged particles in uniform $\vec{E}$-fields, cyclotron motion, and $\vec{E}\times\vec{B}$ drift, showing that the HJ approach recovers all standard results with a unified method.

\section{Hamilton-Jacobi Fundamentals}

\input{concepts/advanced/hj-equation}

\input{concepts/advanced/separation}

\input{concepts/advanced/action-angle}

\input{concepts/advanced/hj-em-coupling}

\section{Mechanics Problems via HJ}

\input{concepts/advanced/free-particle-hj}

\input{concepts/advanced/projectile-hj}

\input{concepts/advanced/sho-hj}

\input{concepts/advanced/kepler-hj}

\input{concepts/advanced/rigid-rotator-hj}

\section{Electromagnetism Problems via HJ}

\input{concepts/advanced/uniform-e-field-hj}

\input{concepts/advanced/cyclotron-hj}

\input{concepts/advanced/crossed-fields-hj}

\input{concepts/advanced/kepler-coulomb-hj}